Applied Bionics and Biomechanics 11 (2014) 39–45DOI 10.3233/ABB-140091IOS Press

39

On two-fluid blood flow through stenosedartery with permeable wall

Rupesh K. Srivastava,b,∗ and V.P. Srivastavaa

aDepartment of Mathematics, Integral University, Lucknow, IndiabDepartment of Mathematics, Ambalika Institute of Management & Technology, Lucknow, India

Abstract. The present investigation concerns the fluid mechanical study on the effects of the permeability of the wall throughan axisymmetric stenosis in an artery assuming that the flowing blood is represented by a two-fluid model. The expressions forthe blood flow characteristics, the impedance, the wall shear stress distribution in the stenotic region and the shearing stress atthe stenosis throat have been derived. Results for the effects of permeability as well as of the peripheral layer on these bloodflow characteristics are quantified through numerical computations and presented graphically and discussed comparatively tovalidate the applicability of the present model.

Keywords: Permeability, darcy number, slip parameter, stenosis throat

1. Introduction

The frequently occurring cardiovascular disease,arteriosclerosis or stenosis, responsible for many ofthe diseases (myocardial infarction, cerebral strokes,angina pectoris, etc.), is the unnatural and abnormalgrowth that develops at various locations of the cardio-vascular system under diseased conditions. Although,the etiology of the initiation of the disease (stenosis) isnot well understood but it is well established that oncethe constriction has developed, it brings about the sig-nificant changes in the flow field (pressure distribution,wall shear stress, impedance, etc.). With the knowledgethat the hemodynamic factors play an important rolein the genesis and the proliferation of arteriosclero-sis, since the first investigation of Mann et al. [13], alarge number of researcher have addressed the stenoticdevelopment problems under various flow situationsincluding Young [27], Young and Tsai [26], Caro et al.

∗Corresponding author: Rupesh K. Srivastav, Department ofMathematics, Integral University, Lucknow, India. E-mail: rupeshk.srivastav@gmail.com.

[3], Shukla et al. [21], Liu et al. [11], Srivastava andcoworkers [22, 23], Mishra et al. [17], Ponalagusamy[19], Layek et al. [10], Tzirtzilakis [25], Mandal et al.(2007), Politis et al. [18], Medhavi [14–16], Srivastavet al. [22, 24], Naturforschung A. (2013), Eldesokyet al. [5–9], and many others.

The flowing blood has been represented by a New-tonian, non-Newtonian, single or double-layered fluidby the investigators in the literature while discussingthe flow through stenoses. It is well known that bloodmay be represented by a single-layered model in largevessel, however, the flow through the small arteriesis known to be a two-layered. Bugliarello and Sevilla[2] and Cokelet [4] have shown experimentally thatfor blood flowing through small vessels, there is acell-free plasma (Newtonian viscous fluid) layer anda core region of suspension of all the erythrocytes. Sri-vastava (2010) concluded that the significance of theperipheral layer increases with decreasing blood vesseldiameter. In addition, the endothelial walls are knownto be highly permeable with ultra microscopic poresthrough which filtration occurs. Cholesterol is believedto increase the permeability of the arterial wall. Such

1176-2322/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved

40 R.K. Srivastav and V.P. Srivastava / On two-fluid blood flow through stenosed artery

increase in permeability results from dilated, damagedor inflamed arterial walls. In view of the discussiongiven above, the research reported here is thereforedevoted to discuss the two-layered blood flow throughan axisymmetric stenosis in an artery with permeablewall. The mathematical model considers the flowingblood as a two-layered Newtonian fluid, consistingof a core region (central layer) of suspension of allthe erythrocytes assumed to be a Newtonian fluid, theviscosity of which may vary depending on the flowconditions and a peripheral region (outer layer) ofanother Newtonian fluid (plasma) of constant viscosity,in an artery with permeable wall.

2. Formulation of the problem

Consider the two-layered axisymmetric flow ofblood through an axisymmetric stenosis, specified atthe location as shown in Fig. 1. The geometry of thestenosis which is assumed to be manifested in the arte-rial wall segment is described as

R(z), R1(z)

R0

= (1, β) − (δ, δ1)

2R0

{1 + cos

2π

L0

(z − d − L0

2

)};

d ≤ z ≤ d + L0

= (1, β), otherwise (1)

where z is the axial coordinate, (R, R1)∼=(R(z), R1(z))are the radius of the (tube, interface) with constric-tion; R0 is the radius of the normal (without stenosis)artery; L0 is the stenosis length, L is the tube lengthand d indicates the location of stenosis, β is the ratio of

Fig. 1. Two-layered blood flow through an axisymmetric stenosiswith permeable wall.

the central core radius to the tube radius in the unob-structed region and (δ, δ1) are the maximum height ofthe stenosis and the bulging of the interface.

The flowing blood is assumed to be represented by atwo-layered Newtonian fluid. The equations describingthe laminar, steady, one-dimensional flow in the caseof a mild stenosis (δ << R0) are expressed [20] as

dp

dz= µp

r

∂

∂r

(r∂

∂r

)up, R1(z) ≤ r ≤ R(z), (2)

dp

dz= µc

r

∂

∂r

(r∂

∂r

)uc, 0 ≤ r ≤ R1(z), (3)

where R1(z) is the radius of the central layer, (up, µp)and (uc, µc) are (velocity, viscosity) of fluid in theperipheral layer (R1(z) ≤ r ≤ R(z)) and central layer(0 ≤ r ≤ R1(z)), respectively; p is the pressure and (r,z) are (radial, axial) coordinates in the two-dimensionalcylindrical polar coordinate system.

The appropriate boundary conditions [1] for thepresent problem may be stated [24] as

∂ uc

∂ r= 0 at r = 0 (4)

up = uc and µp∂ up

∂ r= µc

∂ uc

∂ rat r = R1(z), (5)

up = uB and∂ up

∂ r= α√

k

(uB-uporous

)at r = R(z),

(6)where uporous = − k

µp

dpdz , is the velocity in the perme-

able boundary, uB is the slip velocity, µp is the plasmaviscosity (fluid viscosity in the peripheral layer), kis Darcy number and � (called the slip parameter)is a dimensionless quantity depending on the mate-rial parameters which characterize the structure of thepermeable material within the boundary region.

3. Analysis

The solutions of Eqs. (2) and (3) under the boundaryconditions (4), (5) and (6), yield the expressions forvelocity, up and uc as

R.K. Srivastav and V.P. Srivastava / On two-fluid blood flow through stenosed artery 41

up = − R20

4µp

dp

dz

{(R

R0

)2

−(

r

R0

)2

−2

(R

R0

) ( √k

αR0

)+ 4

k

R20

}, (7)

uc = − R20

4µp

dp

dz

{(R

R0

)2

− µ

(r

R0

)2

−(1−µ)

(R1

R0

)2}

−2

(R

R0

)( √k

αR0

)+ 4

k

R20

, (8)

with µ = µp/µc.

The volumetric flow rate, Q is now calculated as

Q = 2π

⎧⎪⎨⎪⎩

R1∫0

rucdr +R∫

R1

rupdr

⎫⎪⎬⎪⎭

Q = −πR40

8µp

dp

dz

{(R

R0

)4

− (1 − µ)

(R1

R0

)2

+ 8k

R20

(R

R0

)2

− 4√

k

αR0

(R

R0

)3},(9)

Following the argument [21, 23] that the total fluxis equal to the sum of the fluxes across the two regions(peripheral and central), one derives the relations:R1 = � R and �1 = �� (0 ≤ � ≤ 1). An applicationof these relations into the Eq. (9), yields

dp

dz= −8µpQ

πR40

φ(z), (10)

where

φ(z) = 1/{

[ 1 − (1 − µ)β4](R/

R0)4

+8k(R/

R0)2/

R20− 4

√k(R/

R0)3/αR0

}.

The pressure drop, �p(= p at z = 0, - p at z = L)across the stenosis in the tube of length, L is obtainedas

�p =L∫

0

(−dp

dz

)dz

= 8�pQ

πR40

⎧⎨⎩

d∫0

[φ(z)]R/R0=1 dz +d+Lo∫d

φ(z) dz

+L∫

d+Lo

[φ(z)]R/R0=1 dz

⎫⎪⎬⎪⎭ , (11)

The analytical evaluation of the second integral onthe right hand side of Eq. (11) is a formidable taskand therefore shall be evaluated numerically, whereasthe evaluation of first and third integrals are straightforward. Using now the definitions from the publishedliterature [23, 27], one derives the expressions for theimpedance (flow resistance), λ, the wall shear stressdistribution in stenotic region, τw and the shear stressat the stenosis throat, τs in their non-dimensional formas

λ = µ

⎧⎨⎩ (1 - L0/L) η1

η+ η1L0

2π L

2π∫0

ψ(θ)dθ

⎫⎬⎭ (12)

τw = µη1

[1 − (1 − µ)β4](R/

R0)3 + 8k

(R/

R0)/

R20 − 4

√k(R/

R0)2/αR0

, (13)

τs = [τw]R/R0=1−δ/R0 (14)

where

ψ(θ) = [φ(z)]R/R0=a+b cos θ, a = 1 − δ/

2R0,

b = δ/

2R0

η1 = 1 + 8k / R20−4

√k / αR0,

η = 1−(1−µ)β4 + 8k / R20-4

√k / αR0,

λ = λ / λ0,(τw,τs) = (τw,τs) / τ0,

λ0 = 8µcL/η1πR4

0 and τ0 = 4µcQ/η1πR3

0 are theflow resistance and shear stress, respectively for asingle-layered Newtonian fluid in a normal artery (nostenosis) with permeable wall and (λ, τw, τs) are

42 R.K. Srivastav and V.P. Srivastava / On two-fluid blood flow through stenosed artery

respectively, (the impedance, the wall shear stress andthe shearing stress at stenosis throat) obtained from thedefinitions [27]:

λ = �p/

Q, τw = −(R/

2)dp/

dz

and τs = [τw]R/R0=1−δ/R0 .

4. Numerical results and discussion

To discuss the results of the study quantitatively,computer codes are developed to evaluate the ana-lytical result for flow resistance, λ, the wall shearstress, τw, and shear stress at the stenosis throat,τs obtained above in Equations (12)–(14) for vari-ous parameter values and some of the critical resultsare displayed graphically in Figs. 2–13. The variousparameters are selected [1, 24, 27] as: L0(cm) = 1;L(cm) = 1, 2, 5, 10; α = 0.1, 0.2, 0.3, 0.5;

√k =

0, 0.1, 0.2, 0.3, 0.4, 0.5; β = 1, 0.95, 0.90; µ =1, 0.5, 0.3, 0.1; and δ

/R0 = 0, 0.5, 0.10, 0.45,

0.20; etc. It is worth mentioning here that present studycorresponds to impermeable artery case, to single-layered model study, and to no stenosis case forparameter values

√k (here and after called Darcy num-

ber) = 0; β = 1 or µ = 1, and δ/

R0 = 0; respectively.The flow resistance λ, increases with the steno-

sis height, δ/

R0, for any given set of parameters. Atany given stenosis height, δ

/R0, λ decreases with the

peripheral layer viscosity, µ from its maximal magni-tude obtained in a single-layered study (i.e., µ = 1 orβ = 1; Fig. 2).

Fig. 2. Impedance, λ vs. stenosis height δ/Ro for different µ.

Fig. 3. Impedance, λ vs. stenosis height δ/Ro for different α.

Fig. 4. Impedance, λ vs. stenosis height δ/Ro for different κ1/2.

One observes that at any given stenosis height, δ/

R0,the impedance, λ increases with the slip parameter,α (Fig. 3). The blood flow characteristic, λ increaseswith the Darcy number,

√k at any given stenosis

height, δ/

R0 (Fig. 4). The impedance, λ decreases withincreasing tube length L which inturns implies thatλ, increases with increasing value of L0

/L (stenosis

length, Fig. 5).One observes that the flow resistance, λ decreases

rapidly with increasing value of the Darcy number,√k from its maximal magnitude at

√k = 0 (imperme-

able wall) in the range 0 ≤ √k ≤ 0.15 and afterwards

assumes an asymptotic value with increasing valuesof the Darcy number,

√k (Fig. 6). We notice that the

R.K. Srivastav and V.P. Srivastava / On two-fluid blood flow through stenosed artery 43

Fig. 5. Impedance, λ vs. stenosis height δ/Ro for different L.

Fig. 6. Impedance, λ versus Darcy number κ1/2 for different δ/Ro.

blood flow characteristic, λ increases with the slipparameter, α from its minimal magnitude at α = 0.1and approaches to an asymptotic magnitude when αincreases from 0.2 (Fig. 7).

The wall shear in the stenotic region, τw increasesfrom its approached value at z

/L0 = 0 to its peak

value at z/

L0 = 0.5 and then decreases from its peakvalue to its approached value at the end point of theconstriction profile at z

/L0 = 1 for any given set of

parameters (Figs. 8–11). The blood flow characteris-tic, τw decreases with the peripheral layer viscosity, µat any axial location of the constriction profile (Fig. 8).At any point of stenotic region, the wall shear stress,τw increases with Darcy number,

√k (Fig. 9). The flow

characteristic τw also increases with the slip param-

Fig. 7. Impedance, λ versus slip parameter, α for different δ/Ro.

Fig. 8. Wall shear stress in the stenotic region for different µ.

eter, α at any axial location in the stenotic region(Fig. 10).

For any given set of parameters, the wall shear stress,τw increases with the stenosis height, δ

/R0 (Fig. 12).

The blood flow characteristic, τs increases with the slipparameter, α (Fig. 13) for any given set of other param-eters. Numerical results reveal that the variations ofthe shear stress, τs are similar to that of the impedance(flow resistances), λ with respect to any parameter.

5. Conclusions

To observe the effects of the permeability of theartery wall and the peripheral layer on blood flow

44 R.K. Srivastav and V.P. Srivastava / On two-fluid blood flow through stenosed artery

Fig. 9. Wall shear stress in the stenotic region for different κ1/2.

Fig. 10. Wall shear stress in the stenotic region for different α.

Fig. 11. Wall shear stress in the stenotic region for different δ/Ro.

Fig. 12. Shear stress at stenosis throat, τs versus stenosis height,δ/Ro for different µ.

Fig. 13. Shear stress at stenosis throat, τs versus stenosis height,δ/Ro for different α.

characteristics due to the presence of a stenosis, atwo-fluid blood flow of Newtonian fluid through anaxisymmetric stenosis in an artery with permeable wallhas been studied. The study enables one to observe thesimultaneous effects of the wall permeability and theperipheral layer on blood flow characteristics due tothe presence of a stenosis. For any given set of param-eters, the blood flow characteristics (impedance, wallshear stress, etc.) assume lower magnitude in two-fluidmodel than its corresponding value in one-fluid anal-ysis. The impedance decreases with increasing Darcynumber from its maximal magnitude in the case ofimpermeable wall (i.e., at zero Darcy number). It is

R.K. Srivastav and V.P. Srivastava / On two-fluid blood flow through stenosed artery 45

therefore concluded that the existence of permeabilityin the artery wall and the presence of the peripherallayer in the artery helps the functioning of the diseasedartery.

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